# Sh:812

• Shelah, S., Väisänen, P., & Väänänen, J. A. (2005). On ordinals accessible by infinitary languages. Fund. Math., 186(3), 193–214.
• Abstract:
Let \lambda be an infinite cardinal number. The ordinal number \delta(\lambda) is the least ordinal \gamma such that if \phi is any sentence of L_{\lambda^+\omega}, with a unary predicate D and a binary predicate \prec, and \phi has a model M with \langle D^M,\prec^M\rangle a well-ordering of type \ge\gamma, then \phi has a model M' where \langle D^{M'}, \prec^{M'}\rangle is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L_{\lambda^+\omega} is exactly \beth_{\delta(\lambda)}. We show the following theorem.

Theorem Suppose \aleph_0<\lambda<\theta\leq\kappa are cardinal numbers such that \lambda^{<\lambda}=\lambda, {\rm cf}(\theta)\geq \lambda^+ and \mu^\lambda<\theta whenever \mu<\theta, and \kappa^\lambda = \kappa. Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality < \lambda such that in the extension 2^\lambda = \kappa and \delta(\lambda)= \theta.

• published version (22p)
Bib entry
@article{Sh:812,
author = {Shelah, Saharon and V{\"a}is{\"a}nen, Pauli and V{\"a}{\"a}n{\"a}nen, Jouko A.},
title = {{On ordinals accessible by infinitary languages}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {186},
number = {3},
year = {2005},
pages = {193--214},
issn = {0016-2736},
mrnumber = {2191236},
mrclass = {03C75 (03C55 03E35)},
doi = {10.4064/fm186-3-1}
}